Major subspaces of recursively enumerable vector spaces

The main point of this paper is a further development of some aspects of the recent theory of recursively enumerable (r.e.) algebraic structures. Initial work in this area is due to Frölich and Shepherdson [4] and Rabin [10]. Here we are only concerned with vector space structure. The previous work on r.e. vector spaces is due to Dekker [2], [3], Metakides and Nerode [8], Remmel [11], Retzlaff [13], and the author [5].Our object of study is V∞ a countably infinite dimensional fully effective vector space over a countable recursive field . By fully effective we mean that V∞. under a fixed Godel numbering has the following properties:(i) Operations of vector addition and scalar multiplication on V∞ are presented by partial recursive functions on the Gödel numbers of elements of V∞.(ii) V∞ has a dependence algorithm, i.e., there is a uniform effective procedure which applied to any n vectors of V∞ determines whether or not they are linearly independent.We also study , the lattice of r.e. subspaces of V∞ (under the operations of intersection, ⋂ and (weak) sum, +). We note that if is not distributive and is merely modular (see [1]). This fact indicates the essential difference between the lattice of r.e. sets and .

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On r.e. and co-r.e. vector spaces with nonextendible bases

Journal of Symbolic Logic ◽

10.2307/2273351 ◽

1980 ◽

Vol 45 (1) ◽

pp. 20-34 ◽

Cited By ~ 18

Author(s):

J. Remmel

Keyword(s):

Vector Space ◽

Quotient Space ◽

Independent Set ◽

Effective Procedure ◽

Vector Spaces ◽

Natural Numbers ◽

Infinite Dimensional ◽

Recursively Enumerable ◽

Vector Addition ◽

Zero Vector

The concern of this paper is with recursively enumerable and co-recursively enumerable subspaces of a recursively presented vector spaceV∞ over a (finite or infinite) recursive field F which is defined in [6] to consist of a recursive subset U of the natural numbers N and operations of vector addition and scalar multiplication which are partial recursive and under which V∞ becomes a vector space. Throughout this paper, we will identify V∞ with N, say via some fixed Gödel numbering, and assume V∞ is infinite dimensional and has a dependence algorithm, i.e., there is a uniform effective procedure which determines whether or not any given n-tuple v0, …, vn−1 from V∞ is linearly dependent. Various properties of V∞ and its sub-spaces have been studied by Dekker [1], Guhl [3], Metakides and Nerode [6], Kalantari and Retzlaff [4], and the author [7].Given a subspace W of V∞, we say W is r.e. (co-r.e.) if W(V∞ − W) is an r.e. subset of N and write dim(V) for the dimension of V. Given subspaces V, W of V∞, V + W will denote the weak sum of V and W and if V ⋂ M = {0} (where 0 is the zero vector of V∞), we write V ⊕ Winstead of V + W. If W ⊇ V, we write Wmod V for the quotient space. An independent set A ⊆ V∞ is extendible if there is an r.e. independent set I ⊇ A such that I − A is infinite and A is nonextendible if it is not the case An is extendible. A r.e. subspace M ⊇ V∞ is maximal if dim(V∞ mod M) = ∞ and for any r.e. subspace W ⊇ Meither dim(W mod M) < ∞ or dim(V∞ mod W) < ∞.

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Recursive equivalence types of vector spaces

Journal of the Australian Mathematical Society ◽

10.1017/s1446788700022953 ◽

1974 ◽

Vol 18 (3) ◽

pp. 376-384 ◽

Cited By ~ 2

Author(s):

Alan G. Hamilton

Keyword(s):

Vector Space ◽

Vector Spaces ◽

Natural Numbers ◽

Infinite Dimensional ◽

Recursively Enumerable ◽

Countably Infinite

We consider subspaces of a vector space UF, which is countably infinite dimensional over a recursively enumerable field F with recursive operations, where the operations in UF are also recursive, and where, of course, F and UF are sets of natural numbers. It is the object of this paper to investigate recursive equivalence types of such vector spaces and the ways in which their properties are analogous to and depend on properties of recursive equivalence types of sets.

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Maximal and Cohesive vector spaces

Journal of Symbolic Logic ◽

10.2307/2272868 ◽

1977 ◽

Vol 42 (3) ◽

pp. 400-418 ◽

Cited By ~ 12

Author(s):

J. B. Remmel

Keyword(s):

Vector Space ◽

Quotient Space ◽

Effective Procedure ◽

Vector Spaces ◽

Recursively Enumerable Set ◽

Infinite Dimensional ◽

Recursively Enumerable ◽

Vector Addition ◽

Zero Vector ◽

Recursive Set

Let N denote the natural numbers. If A ⊆ N, we write Ā for the complement of A in N. A set A ⊆ N is cohesive if (i) A is infinite and (ii) for any recursively enumerable set W either W ∩ A or ∩ A is finite. A r.e. set M ⊆ N is maximal if is cohesive.A recursively presented vector space (r.p.v.s.) U over a recursive field F consists of a recursive set U ⊆ N and operations of vector addition and scalar multiplication which are partial recursive and under which U becomes a vector space. A r.p.v.s. U has a dependence algorithm if there is a uniform effective procedure which applied to any n-tuple ν0, ν1, …, νn−1 of elements of U determines whether or not ν0, ν1 …, νn−1 are linearly dependent. Throughout this paper we assume that if U is a r.p.v.s. over a recursive field F then U is infinite dimensional and U = N. If W ⊆ U, then we say W is recursive (r.e., etc.) iff W is a recursive (r.e., etc.) subset of N. If S ⊆ U, we write (S)* for the subspace generated by S. If V1 and V2 are subspaces of U such that V1 ∩ V2 ={} (where is the zero vector of U), then we write V1 ⊕ V2 for (V1 ∪ V2)*. If V1 ⊆ V2⊆U are subspaces, we write V2/V1 for the quotient space.

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Bases and α-dimensions of countable vector spaces with recursive operations

Journal of Symbolic Logic ◽

10.1017/s0022481200092276 ◽

1970 ◽

Vol 35 (1) ◽

pp. 85-96

Author(s):

Alan G. Hamilton

Keyword(s):

Vector Space ◽

Vector Spaces ◽

Natural Numbers ◽

Infinite Dimensional ◽

Recursively Enumerable ◽

Countably Infinite

This paper is based on the notions originally described by Dekker [2], [3], and the reader is referred to these for explanation of notation etc. Briefly, we are concerned with a countably infinite dimensional countable vector space Ū with recursive operations, regarded as being coded as a set of natural numbers. Necessarily, then, Ū must be a vector space over a field which itself is in some sense recursively enumerable and has recursive operations.

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Simple and hyperhypersimple vector spaces

Journal of Symbolic Logic ◽

10.2307/2272824 ◽

1978 ◽

Vol 43 (2) ◽

pp. 260-269 ◽

Cited By ~ 9

Author(s):

Allen Retzlaff

Keyword(s):

Vector Space ◽

Modular Lattice ◽

Final Section ◽

Early Research ◽

Vector Spaces ◽

Dimensional Vector ◽

Dimensional Vector Space ◽

Infinite Dimensional ◽

Recursively Enumerable ◽

Algebraic Properties

AbstractLet V∞ be a fixed, fully effective, infinite dimensional vector space. Let be the lattice consisting of the recursively enumerable (r.e.) subspaces of V∞, under the operations of intersection and weak sum (see §1 for precise definitions). In this article we examine the algebraic properties of .Early research on recursively enumerable algebraic structures was done by Rabin [14], Frölich and Shepherdson [5], Dekker [3], Hamilton [7], and Guhl [6]. Our results are based upon the more recent work concerning vector spaces of Metakides and Nerode [12], Crossley and Nerode [2], Remmel [15], [16], and Kalantari [8].In the main theorem below, we extend a result of Lachlan from the lattice of r.e. sets to . We define hyperhypersimple vector spaces, discuss some of their properties and show if A, B ∈ , and A is a hyperhypersimple subspace of B then there is a recursive space C such that A + C = B. It will be proven that if V ∈ and the lattice of superspaces of V is a complemented modular lattice then V is hyperhypersimple. The final section contains a summary of related results concerning maximality and simplicity.

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FINITE-DIMENSIONAL ORDERED VECTOR SPACES WITH RIESZ INTERPOLATION AND EFFROS–SHEN’S UNIMODULARITY CONJECTURE

Journal of the Australian Mathematical Society ◽

10.1017/s1446788716000033 ◽

2016 ◽

Vol 101 (2) ◽

pp. 277-287

Author(s):

AARON TIKUISIS

Keyword(s):

Vector Space ◽

Inductive Limit ◽

Space Structure ◽

Vector Spaces ◽

Ordered Vector Space ◽

Finite Dimensional ◽

Ordered Vector Spaces

It is shown that, for any field $\mathbb{F}\subseteq \mathbb{R}$, any ordered vector space structure of $\mathbb{F}^{n}$ with Riesz interpolation is given by an inductive limit of a sequence with finite stages $(\mathbb{F}^{n},\mathbb{F}_{\geq 0}^{n})$ (where $n$ does not change). This relates to a conjecture of Effros and Shen, since disproven, which is given by the same statement, except with $\mathbb{F}$ replaced by the integers, $\mathbb{Z}$. Indeed, it shows that although Effros and Shen’s conjecture is false, it is true after tensoring with $\mathbb{Q}$.

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Isomorphisms on countable vector spaces with recursive operations

Journal of the Australian Mathematical Society ◽

10.1017/s1446788700019960 ◽

1974 ◽

Vol 18 (2) ◽

pp. 230-235 ◽

Cited By ~ 2

Author(s):

Robert I. Soare

Keyword(s):

Vector Space ◽

Independent Set ◽

Vector Spaces ◽

Recursive Structure ◽

Recursively Enumerable ◽

Linearly Independent ◽

Vector Addition

Terminology and notation may be found in Dekker [1] and [2]. Briefly, we fix a recursively enumerable (r.e.) field F with recursive structure, and let Ū be the vector space over F consisting of ultimately vanishing countable sequences of elements of F with the usual definitions of vector addition and multiplication by a scalar. A subspace V of Ū is called an α-space if V has a basis B which is contained in some r.e. linearly independent set S.

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Controlling the dependence degree of a recursively enumerable vector space

Journal of Symbolic Logic ◽

10.2307/2271945 ◽

1978 ◽

Vol 43 (1) ◽

pp. 13-22 ◽

Cited By ~ 18

Author(s):

Richard A. Shore

Keyword(s):

Vector Space ◽

Topological Structure ◽

Independent Set ◽

Original Data ◽

Recursion Theory ◽

Vector Spaces ◽

Additional Structure ◽

Recursively Enumerable ◽

Added Benefit ◽

And Algebra

Early work combining recursion theory and algebra had (at least) two different sets of motivations. First the precise setting of recursion theory offered a chance to make formal classical concerns as to the effective or algorithmic nature of algebraic constructions. As an added benefit the formalization gives one the opportunity of proving that certain constructions cannot be done effectively even when the original data is presented in a recursive way. One important example of this sort of approach is the work of Frohlich and Shepardson [1955] in field theory. Another motivation for the introduction of recursion theory to algebra is given by Rabin [1960]. One hopes to mathematically enrich algebra by the additional structure provided by the notion of computability much as topological structure enriches group theory. Another example of this sort is provided in Dekker [1969] and [1971] where the added structure is that of recursive equivalence types. (This particular structural view culminates in the monograph of Crossley and Nerode [1974].)More recently there is the work of Metakides and Nerode [1975], [1977] which combines both approaches. Thus, for example, working with vector spaces they show in a very strong way that one cannot always effectively extend a given (even recursive) independent set to a basis for a (recursive) vector space.

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Products of idempotent endomorphisms of an independence algebra of infinite rank

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100071607 ◽

1993 ◽

Vol 114 (2) ◽

pp. 303-319 ◽

Cited By ~ 27

Author(s):

John Fountain ◽

Andrew Lewin

Keyword(s):

Vector Space ◽

Vector Spaces ◽

Dimensional Vector ◽

Dimensional Vector Space ◽

Infinite Dimensional ◽

Common Generalization ◽

Finite Dimensional Vector Space ◽

Infinite Rank ◽

Finite Dimensional ◽

Linear Maps

AbstractIn 1966, J. M. Howie characterized the self-maps of a set which can be written as a product (under composition) of idempotent self-maps of the same set. In 1967, J. A. Erdos considered the analogous question for linear maps of a finite dimensional vector space and in 1985, Reynolds and Sullivan solved the problem for linear maps of an infinite dimensional vector space. Using the concept of independence algebra, the authors gave a common generalization of the results of Howie and Erdos for the cases of finite sets and finite dimensional vector spaces. In the present paper we introduce strong independence algebras and provide a common generalization of the results of Howie and Reynolds and Sullivan for the cases of infinite sets and infinite dimensional vector spaces.

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